Mastering Quadratic Equations: Step-by-Step Solutions

Oct 16, 2025 by ADMIN 54 views

Hey math enthusiasts! Ready to dive into the world of quadratic equations? This guide is designed to help you conquer those problems step-by-step. We'll break down each equation, providing clear explanations and solutions. So, grab your pencils, and let's get started!

Understanding the Basics: Expanding Binomials

Before we jump into the problems, let's quickly recap how to expand binomials. Remember, when you multiply two binomials (expressions with two terms), you'll often use the FOIL method. FOIL stands for First, Outer, Inner, Last. Here’s a quick breakdown:

First: Multiply the first terms of each binomial.
Outer: Multiply the outer terms of the binomials.
Inner: Multiply the inner terms of the binomials.
Last: Multiply the last terms of each binomial.

For example, to expand (x + 2)(x + 3), you'd do the following:

First: x * x = x²
Outer: x * 3 = 3x
Inner: 2 * x = 2x
Last: 2 * 3 = 6

Now, combine these terms: x² + 3x + 2x + 6. Finally, simplify by combining like terms: x² + 5x + 6.

Now, with this foundation in place, let's solve some problems. Remember to keep the FOIL method in mind as we work through the equations. By understanding and consistently applying these rules, you will be well on your way to mastering these equations. The best way to learn math, is of course, by practicing a lot. So, do not hesitate to revisit these problems and work through them again.

Solving the Equations: Detailed Solutions

Now let's tackle the given set of quadratic equations, breaking down each one into easily digestible steps. Pay attention to how the FOIL method is implemented in each solution. We will use the proper order to solve these problems.

a) (x + 1)(x + 2)

First: x * x = x²
Outer: x * 2 = 2x
Inner: 1 * x = x
Last: 1 * 2 = 2

Combining terms: x² + 2x + x + 2 Simplifying: x² + 3x + 2

b) (x - 1)(x - 2)

First: x * x = x²
Outer: x * -2 = -2x
Inner: -1 * x = -x
Last: -1 * -2 = 2

Combining terms: x² - 2x - x + 2 Simplifying: x² - 3x + 2

c) (x + 3)(x + 4)

First: x * x = x²
Outer: x * 4 = 4x
Inner: 3 * x = 3x
Last: 3 * 4 = 12

Combining terms: x² + 4x + 3x + 12 Simplifying: x² + 7x + 12

d) (x - 3)(x - 4)

First: x * x = x²
Outer: x * -4 = -4x
Inner: -3 * x = -3x
Last: -3 * -4 = 12

Combining terms: x² - 4x - 3x + 12 Simplifying: x² - 7x + 12

e) (x + 2)(x - 6)

First: x * x = x²
Outer: x * -6 = -6x
Inner: 2 * x = 2x
Last: 2 * -6 = -12

Combining terms: x² - 6x + 2x - 12 Simplifying: x² - 4x - 12

f) (x - 2)(x + 6)

First: x * x = x²
Outer: x * 6 = 6x
Inner: -2 * x = -2x
Last: -2 * 6 = -12

Combining terms: x² + 6x - 2x - 12 Simplifying: x² + 4x - 12

g) (x + 5)(x + 8)

First: x * x = x²
Outer: x * 8 = 8x
Inner: 5 * x = 5x
Last: 5 * 8 = 40

Combining terms: x² + 8x + 5x + 40 Simplifying: x² + 13x + 40

h) (5x + 1)(8x + 1)

First: 5x * 8x = 40x²
Outer: 5x * 1 = 5x
Inner: 1 * 8x = 8x
Last: 1 * 1 = 1

Combining terms: 40x² + 5x + 8x + 1 Simplifying: 40x² + 13x + 1

i) (x + 2)(2x + 1)

First: x * 2x = 2x²
Outer: x * 1 = x
Inner: 2 * 2x = 4x
Last: 2 * 1 = 2

Combining terms: 2x² + x + 4x + 2 Simplifying: 2x² + 5x + 2

j) (3x - 1)(x - 3)

First: 3x * x = 3x²
Outer: 3x * -3 = -9x
Inner: -1 * x = -x
Last: -1 * -3 = 3

Combining terms: 3x² - 9x - x + 3 Simplifying: 3x² - 10x + 3

k) (4x - 3)(2x - 5)

First: 4x * 2x = 8x²
Outer: 4x * -5 = -20x
Inner: -3 * 2x = -6x
Last: -3 * -5 = 15

Combining terms: 8x² - 20x - 6x + 15 Simplifying: 8x² - 26x + 15

l) (2x + 7)(3x - 8)

First: 2x * 3x = 6x²
Outer: 2x * -8 = -16x
Inner: 7 * 3x = 21x
Last: 7 * -8 = -56

Combining terms: 6x² - 16x + 21x - 56 Simplifying: 6x² + 5x - 56

Tips and Tricks for Success

To become truly proficient in solving these types of equations, consistent practice is key. Try these additional strategies to strengthen your understanding:

Practice Regularly: Work through various problems every day. The more you solve, the more comfortable you'll become. Set aside dedicated time to work on equations. Even a few minutes each day can make a significant difference. Regular practice helps solidify your understanding and improves your problem-solving speed.
Check Your Work: Always verify your answers. Substitute the solutions back into the original equation to ensure they are correct. A simple mistake can lead to incorrect results. Take your time and double-check each step. This habit helps catch errors and reinforces your understanding.
Understand the Concepts: Don't just memorize the steps. Understand why each step works. This deeper understanding will help you solve more complex problems. Explore the mathematical principles behind the methods you use. Understanding the "why" behind the "how" builds a stronger foundation for solving problems.
Ask for Help: Don't hesitate to seek help from teachers, tutors, or online resources. Get clarification when you're stuck. If you're struggling with a concept, ask for help immediately. Teachers, tutors, and online forums are great resources for getting assistance and gaining insights.
Review Your Mistakes: Analyze your errors to learn from them. Understanding where you went wrong is as important as getting the correct answer. Reviewing your mistakes helps identify recurring patterns and areas for improvement. Every mistake is a learning opportunity.

Conclusion: Your Next Steps

Great job working through these problems! You've taken the first step toward mastering quadratic equations. Remember, the journey to mathematical proficiency involves consistent effort and a willingness to learn. Keep practicing, reviewing, and asking questions. With each problem you solve, you'll gain confidence and sharpen your skills. Continue to explore different types of problems and solutions, and your understanding will continue to grow.

Keep up the great work, and happy calculating!